From what I know, the Prophet of the LDS Church holds the position of Seer, as Joseph Smith did. If this is so, what specific privileges and powers come with that position? Does this include the ability to translate ancient languages as it did for Joseph Smith?

2 Answers
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13 Now Ammon said unto him: I can assuredly tell thee, O king, of a
man that can translate the records; for he has wherewith that he can
look, and translate all records that are of ancient date; and it is a
gift from God. And the things are called interpreters, and no man can
look in them except he be commanded, lest he should look for that he
ought not and he should perish. And whosoever is commanded to look in
them, the same is called seer.

14 And behold, the king of the people who are in the land of
Zarahemla is the man that is commanded to do these things, and who has
this high gift from God.

15 And the king said that a seer is greater than a prophet.

16 And Ammon said that a seer is a revelator and a prophet also; and
a gift which is greater can no man have, except he should possess the
power of God, which no man can; yet a man may have great power given
him from God.

17 But a seer can know of things which are past, and also of things
which are to come, and by them shall all things be revealed, or,
rather, shall secret things be made manifest, and hidden things shall
come to light, and things which are not known shall be made known by
them, and also things shall be made known by them which otherwise
could not be known.

18 Thus God has provided a means that man, through faith, might work
mighty miracles; therefore he becometh a great benefit to his fellow
beings.

Further, President Benson, president of the Quorum of the Twelve Apostles in 1980, spoke about 14 fundamentals to know about following a prophet. In that speech, he declared some of the authority and rights the prophet has while he maintains his office. I've summarized them here but they all demand further explanation and detail (because, I believe, many of these claims are very bold and forward) -- I recommend an in-depth study of the full article and its references.

The prophet is the only man who speaks for the Lord in everything.

The living prophet is more vital to us than the standard works.

The living prophet is more important to us than a dead prophet.

The prophet will never lead the Church astray.

The prophet is not required to have any particular earthly training or credentials to speak on any subject or act on any matter at any
time.

The prophet does not have to say ‘Thus saith the Lord’ to give us scripture.

The prophet tells us what we need to know, not always what we want to know

The prophet is not limited by men’s reasoning.

The prophet can receive revelation on any matter—temporal or spiritual.

The prophet may be involved in civic matters.

The two groups who have the greatest difficulty in following the prophet are the proud who are learned and the proud who are rich.

The prophet will not necessarily be popular with the world or the worldly.

The prophet and his counselors make up the First Presidency—the highest quorum in the Church.

The prophet and the presidency—the living prophet and the first presidency—follow them and be blessed; reject them and suffer.

+1 for use of references that are -- at least within the Mormon tradition -- considered authoritative, and for linking to a site (BYU) that -- again, within that tradition -- is considered reliable.
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Philip SchaffJul 24 '12 at 18:27

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One correction: Ezra Taft Benson was not president of the church until 1985. The linked talk was given while he was president of the Quorum of the Twelve Apostles.
–
Fred LarsonNov 13 '14 at 18:20

13 Now Ammon said unto him: I can assuredly tell thee, O king, of a
man that can translate the records; for he has wherewith that he can
look, and translate all records that are of ancient date; and it is a
gift from God. And the things are called interpreters, and no man can
look in them except he be commanded, lest he should look for that he
ought not and he should perish. And whosoever is commanded to look in
them, the same is called seer.

14 And behold, the king of the people who are in the land of
Zarahemla is the man that is commanded to do these things, and who has
this high gift from God.

15 And the king said that a seer is greater than a prophet.

16 And Ammon said that a seer is a revelator and a prophet also; and
a gift which is greater can no man have, except he should possess the
power of God, which no man can; yet a man may have great power given
him from God.

17 But a seer can know of things which are past, and also of things
which are to come, and by them shall all things be revealed, or,
rather, shall secret things be made manifest, and hidden things shall
come to light, and things which are not known shall be made known by
them, and also things shall be made known by them which otherwise
could not be known.

18 Thus God has provided a means that man, through faith, might work
mighty miracles; therefore he becometh a great benefit to his fellow
beings.

So we see that a seer is a person who has power and authority from God to reveal that which is not known, including the translation of ancient records, but also knowledge "of things which are past and also of things which are to come," and that this power works through faith.

This is an important point when discussing the hypothetical abilities granted by God to such a person, because in the Book of Mormon we find an explanation of faith that closely parallels Paul's definition from Hebrews 11, but with one important clarification (emphasis added):

21 And now as I said concerning faith—faith is not to have a perfect knowledge of things; therefore if ye have faith ye hope for things which are not seen, which are true.

So faith is only valid when it is based off of something that is actually true and correct. This means that a seer would only be able to use the gifts that God has granted to him for purposes that conform to God's plan, and not for arbitrary or personal use.